Question

How many different words can be formed using all the letters of the word TUESDAY such that no two vowels are together?

• A. 720
• B. 120
• C. 360
• D. 1440

Hint:

Gap Method shortcut: If you have $C$ consonants and $V$ vowels, and no two vowels can be together, the formula is $C! \times \, ^{C+1}P_V$.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
When "no two" items are together, we use the Gap Method. We first arrange the items that have no constraints (consonants) and then place the constrained items (vowels) into the gaps created between them.

Step 2: Key Formula or Approach:
1. Arrange $n$ consonants.
2. Identify $n+1$ gaps.
3. Total = (Consonant arrangements) $\times$ (Ways to pick and arrange gaps for vowels).

Step 3: Detailed Explanation:
1. Word: TUESDAY. Consonants (4): T, S, D, Y. Vowels (3): U, E, A. 2. Arrange consonants: $4! = 24$. 3. Gaps created: _ T _ S _ D _ Y _ (Total 5 gaps). 4. Select 3 gaps for the 3 vowels and arrange them: $^5P_3$. \[ ^5P_3 = 5 \times 4 \times 3 = 60 \] 5. Total arrangements = $24 \times 60 = 1440$.

Step 4: Final Answer:
The number of different words is 1440.